2010/06/10 by Wei Dai, Dai, Wei, Ely Kerman +3 · 4 citations
Computer Science · Engineering · Mathematics · #Advanced Image Processing Techniques #Algorithm #Closure (psychology) #Combinatorics #Computer science #Function (biology) #Low-rank approximation #Mathematical optimization #Mathematics #Matrix (chemical analysis) #Matrix completion #Matrix function #Metric (unit) #Pure mathematics #Rank (graph theory) #Set (abstract data type) #Sparse and Compressive Sensing Techniques #Symmetric matrix #Synthetic Aperture Radar (SAR) Applications and Techniques
paper · pdf · doi:10.48550/arxiv.1006.2086
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2010/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The low-rank matrix completion problem can be succinctly stated as follows: given a subset of the entries of a matrix, find a low-rank matrix consistent with the observations. While several low-complexity algorithms for matrix completion have been proposed so far, it remains an open problem to devise search procedures with provable performance guarantees for a broad class of matrix models. The standard approach to the problem, which involves the minimization of an objective function defined using the Frobenius metric, has inherent difficulties: the objective function is not continuous and the solution set is not closed. To address this problem, we consider an optimization procedure that searches for a column (or row) space that is geometrically consistent with the partial observations. The geometric objective function is continuous everywhere and the solution set is the closure of the solution set of the Frobenius metric. We also preclude the existence of local minimizers, and hence establish strong performance guarantees, for special completion scenarios, which do not require matrix incoherence or large matrix size.